Wyluli --
Don't use the red bulb solder suckers from rat shack... they don't work well. I use the spring loaded (pen shaped) one from bob roberts. I like it a lot.
With old caps, I usually heat one end up and rock it one way, then heat the other and rock the other and then it's usually one more touch and I can pull it out. Remember to never push back into the board as you could raise the trace as the solder hardens
To remove the excess, use the sucker first before removing the component, then if any is left you can try sucking from the component side with the spring loaded one, or sometimes I've had to push a needle into the hole (from the bottom to not break the trace) while heating, then twist slowly as it dries and then pull out and it should be open enough to get a new lead through.
Generally there is no static sensitive IC's on the boards unless its a really new or advanced board. Most of the other components can take small zapps on the old boards. However, before handling the board, even after it's been removed from the chassis, be sure the capacitors are discharged...especially the largest physical capacitor on the board..in the power section for power stabilization. That cap can hold a mighty zap that can hurt you touching it, and can make a nice loud zap if shorted.
This usually happend when you have started working on a board that is bad, like a blown fuse or HOT that gave no picture. I had a G07 almost bite me, and instead zapped a plier I accidentially laid across the bottom of the circuit board... scared the heck out of me!
I've never used braids to remove solder.. dunno how they work.